December 7, 2012
Location: Janskerkhof 15a (Utrecht), room 101
Mappings of the double two-dimensional Ising model
Starting from two independent Ising models, one can construct a new model: the XOR Ising model. In the first part of the talk, we will show how loops of the XOR Ising model can be mapped to loops of a bipartite dimer model. This allows us to partially solve a conjecture of Wilson, which states that, at criticality, loops of the XOR Ising model are level lines of a Gaussian free field. In the second part of the talk, we will show how, at criticality, the double Ising model can be mapped to spanning trees, thus showing an unexpected relation between two models of statistical mechanics.
The first part of the talk is based on joint work with Cédric Boutillier.
Stochastic representations of quantum Gibbs states II: Transverse field Ising models
The talk will be devoted to two stochastic representations of the quantum transverse field Ising model:
1. Random cluster representation, which is a space-time generalization of the usual FK-percolation. In the case of mean field interactions such representation leads to a natural space-time version of the Erdős–Rényi random graph. In this way the classical Erdős–Rényi critical inverse temperature becomes the end-point of the critical curve, which can be exactly computed (joint work with Anna Levit).
2. Random current representation, which is a space-time version of the classical random current representation of GHS and Aizenman. This part will be based on a recent joint work with Nick Crawford. A crucial quantum switching lemma enables an adaptation of the classical approach to proving the sharpness of phase transition as developed by Aizenman, Barsky and Fernández.